Spherical solid particles containing B are roasted at constant temperature in an oven by gas of constant composition. Solids are converted to give firm non-flaking product according to the shrinking core model. From the following conversion data determine the rate controlling mechanism for the transformation of solid.

$$\begin{array}{|c|c|c|} \hline \mathrm{{d p}, {m m}} & \mathrm{X}_{\mathrm{B}} & t,s \\ \hline 1 & 1 & 200 \\ \hline 1.5 & 1 & 450 \\ \hline \end{array}$$

Solution :

Here conversion in case of both spherical particles is complete ($\mathrm {X_B}=1$) so the time given is $\tau $

For $\mathrm {dp_1=1~mm, \quad \tau_1=t_1=200~s}$

For $\mathrm{ dp_1=1.5~mm, \quad \tau_2=t_2=450~s}$

1) Film diffusion controls :

We have $\mathrm {\tau \propto R}$

$\mathrm {r_{1} \propto R_{1}}$ and $\tau_{2} \propto \mathrm R_{2}$

$$\begin {aligned}\frac{\tau_{1}}{\tau_{2}} &=\frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\frac{(\mathrm{dp}_1)}{\left(\mathrm{~dp}_{2}\right)}\\ \mathrm {LHS }:~\quad \frac{\tau_{1}}{\tau_{2}}&=\frac{200}{450}=0.444 \\ \mathrm {RHS }:\quad \frac{R_{1}}{R_{2}}&=\frac{d p_{1}}{d p_{2}}=\frac{1}{1.5}=0.667 \end {aligned}$$

LHS not equal to RHS

$\therefore$ Film diffusion does not control the rate.

2) Ash diffusion controls :

We have $\tau \propto \mathrm{R}^{2}$

$\tau_{1} \propto \mathrm{R}_{1}^{2}$ and $\tau_{2} \propto \mathrm{R}_{2}^{2}$

$$\begin {aligned}\frac{\tau_{1}}{\tau_{2}} &=\frac{\mathrm{R}_{1}^{2}}{\mathrm{R}_{2}^{2}}=\frac{(\mathrm{dp_1})^{2}}{\left(\mathrm{~dp}_{2}\right)^{2}} \\ \mathrm{LHS : } \quad ~\frac{\tau_{1}}{\tau_{2}} &=\frac{200}{450}=0.444 \\ \mathrm{ RHS : }\quad \frac{\mathrm R_{1}^{2}}{\mathrm R_{2}^{2}} &=\frac{\left(\mathrm {d p_{1}}\right)^{2}}{\left(\mathrm {d p_{2}}\right)^{2}}=\frac{(1)^{2}}{(1.5)^{2}}=0.444 \end{aligned}$$

LHS = RHS

Ash diffusion controls the rate of transformation of solid.

3) Chemical reaction controls :

We have again $\tau \propto \mathrm{R}$ as that with film diffusion obtain the same results as that for film diffusion.

So chemical reaction does not control the rate.